You could have a Close Call (a strange encounter that indicates you are in the right place and at the right level!)

All of the above consume 1 AP.

The level of the monsters you encounter is determined by your Level and the Level Range of the Place you are exploring. Provided that your Level is within the Range you will explore at that level and the monsters you encounter will be that level. After each fight you have the option to move up one to a tougher level or down one to a safer level (so long as that is still within the Level Range). The higher the level the better the loot, and vice versa.